The present invention relates generally to a method and apparatus for the multivariate allocation of resources. In particular, the present invention provides a method and apparatus for modeling objects, such as customers and suppliers, and thereafter presents a method for solving a resulting multivariate expected value function as a closed form expression.
According to microeconomic theory, a recurring problem of large scale manufacturing processes is the allocation of valuable resources to meet uncertain consumer demand over a large number of products. In the most general scenario, certain resources are shared among many products. As a result, depletion of any one resource by a product demanding a high amount of that resource will preclude the manufacture of all other products requiring that same resource for manufacture.
A simple solution to this problem would be to maintain a large inventory of all relevant resources. This, however, is not an effective solution because resource inventory accrues a cost to the company. Some fast moving, or volatile inventory materials might decrease in value at an exponential rate. Certain types of memory components, for example, are known to depreciate at a rate of approximately one percent per week. If significant inventories are maintained for a long period of time, then such components will lose most their value before being used. Sometimes such components can even become valueless. This adds unnecessary costs to the manufacture of the product, and ultimately the price offered to the consumer. If such costs cannot be passed onto the consumer, as is typical in competitive markets, then such costs will come directly out of a company""s profits.
A converse solution would be to maintain low inventories, and then procure the parts from the suppliers on an as-needed basis. This is not an effective solution because procuring scarce parts on a short-term basis often carries added costs, or xe2x80x9cpenalty costs.xe2x80x9d For instance, parts that are ordered during the normal course of business carry a certain cost. Parts that are required on an expedited basis are often priced at higher levels. These costs are usually ratcheted upwards (on a lock-step basis, or otherwise) as the demand for product increases. Hence, if a significant number of parts are needed to complete the manufacture of a series of products, then a significant premium will have to be paid to the suppliers in order to procure sufficient parts. As a worst case, such scarce parts might not be available at any price. If the parts cannot be procured, then the end products cannot be manufactured. This will obviously result in lost sales. Significant lost sales can even lead to overall lost market share and reduced customer loyalty.
Accordingly, the general solution to such problems involves finding the allocation of components (or resources) that maximizes value (i.e. profits, or revenues minus costs) across the set of products (or refinements) to be manufactured. More importantly, the solution must take into account the xe2x80x9chorizontalxe2x80x9d interaction effects among products, as well as the xe2x80x9cverticalxe2x80x9d consumption effects between products and components.
Simple prior art solutions to allocation problems include Manufacturing Requirements Primer (MRP) models. The basic principle behind an MRP model is to formulate a xe2x80x9crecipexe2x80x9d pertaining to the manufacture of a product, i.e. one microprocessor, two memory modules, and one storage device might be used to make up an end product. An MRP model performs a count of such components and tallies them up across the number (and type) of desired end products. Thereafter the MRP system schedules the allocation and delivery of such components at the factory so that the manufactured products come out on time, and in the proper order. However, such MRP models and solutions do not adequately account for the interactive effects among products and components. Moreover problematic, MRP models and solutions typically assume fixed, known demands on products.
Other prior solutions have been proposed which partially address the horizontal interaction effects and the vertical consumption effects, with the result being an expected value function which must be solved for a given value. The expected value function is generally the expectation of a linear, or polynomial, or exponential function over a multivariate normal (or other type) distribution. The more interactions that occur between the various components of a model, the higher the order of the expected value function. For any model involving a plurality of interactions, the form of this expression usually becomes a very complicated multivariate integral. To solve this function over a plurality of variables, prior solutions must employ significant computer resources. Often the best approach in solving such integrals involves applying a xe2x80x9cMonte Carloxe2x80x9d technique, which in the end serves as only an approximation of a result. Monte Carlo techniques also takes massive amounts of computer processing power (i.e. a supercomputer) to solve, and cannot generally be solved in a reasonable period of time.
Given that the solution to such allocation problems often carries significant financial ramifications for a company, it is important to produce a solution which is more than just an estimate. Moreover, an expression is needed which can be solved in a reasonable amount of time, and without super-computer resources. Hence, a modeling technique is needed that will properly account for the horizontal and vertical interactions between certain modeled elements. A solution technique is thereafter needed which will present a closed form expression of the resulting function, wherein it will not be necessary to solve multiple integrals in order to determine a solution. This closed formed expression should also be executable on ordinary computer resources, and in a reasonable period of time, despite the multivariate nature of the problem.
To achieve the foregoing, and in accordance with the purpose of the present invention, a method and apparatus are disclosed that provides an efficient solution for the multivariate allocation of resources.
The theory and solution generalizes to any model of resource consumption, in relation to producing a xe2x80x9crefinement.xe2x80x9d The term xe2x80x9crefinement,xe2x80x9d as used through this document, is generally intended to represent an end result (i.e. product) which might be produced from a set of resources (i.e. components, or the like). Therefore, a typical refinement-resource framework might involve product-component models, wherein certain components are used to comprise certain products. Resources might also include available liquid capital for investment, bonds, stocks, and options. The present system might also be used to consider assets, a portfolio of assets, or consumption of those assets, such as energy (e.g. gas, nuclear, electric), space, real estate, etc. Another example problem includes the allocation of manpower. For instance, in association with manpower problems, a set of resources (i.e. employees) exists that might be used by many different end sources (i.e. work tasks, or jobs). Sharing of such manpower will involve a complex mix of parameters. In order to maximize productivity (or revenue), the assets (or resources) will need to be analyzed, in relation to production (or refinement) goals.
As yet another example, a company might entertain a portfolio of development projects. Each project requires the allocation of capital, work force, and new equipment. Certain factors remain uncertain, including project growth as based upon the success of the venture, market indicators of the interest in the product, uncertain market pressures and demand, and the synergization and cannibalization offered by competing projects. The company desires to know how to best allocate its resources over the various projects in order to maximize revenues in the face of the aforementioned uncertainties.
According to one aspect of the present invention, relevant models and associated equations are formulated, wherein the equations are solved for certain values (or ranges). The models might consist of a set of resources (e.g. components) and a set of refinements of those resources (e.g. products). The resource consumption is based on a linear relationship between each refinement and its set of supporting resources (e.g. the bill of materials for a product). Each resource is typically shared among several refinements. There is a demand distribution for the refinements that is a multivariate normal distribution (e.g. future product demand for next quarter, or the like). There is also a value function that is a linear, polynomial, or exponential function of the refinement demands and other associated parameters of the model. For instance, the value function might include a revenue function for certain products, and be expressed as a sum of the products of the margin and demand for each refinement (or product). Of interest to any company, analyst, or the like, is the computation of the statistical expectation of the value, function at a given resource allocation, and for a given multivariate normally distributed demand profile. This is referred to as the expected value function.
According to another aspect of the present invention, this expected value function is transformed into a closed form expression. According to the solution offered by the present method and apparatus, each resource, and the refinements that it supports, generates a resource hyperplane in the demand space such that on one half of the hyperplane, the resource is in excess of the combined demand generated by the refinements. On the other half of the hyperplane, there is an insufficient amount of the resource to meet the combined refinement demand.
The complete set of refinements generates an intersecting set of hyperplanes in the demand space such that the joint intersection forms a polytope on which resource allocation fulfills refinement demand. However, because the resource is in excess, it generates a resource loss in the value function, also known as erosion. In the complement space of the polytope, there can be also be resource losses, i.e. certain components erode because they were not fully consumed, given the exhaustion of another key component. There can also be refinement losses in the form of refinement demands that were not fully met.
The particular form of the value function might also depend on certain policies (i.e. business, strategic, etc.) associated with allocating resources to the refinements. The expected value function might then be solved under various policies, including: a priority policy that fulfills refinement demands in some pre-specified rank order; and a uniform policy that uniformly meets refinement demands.
A sequence of three linear transformations are used to reduce the expected value function to a multivariate polynomial function of single variable integrals, wherein each such integral has a closed form expression. The expected value function can therefore be reduced to a closed-form expression that depends upon (among other things) the resource allocations, the coefficients of the linear combination of resources for each refinement, and the coefficients of the value function.
The first linear transformation uses a Cholesky decomposition of the covariance matrix and thereby reduces the mean and covariance matrix of the refinement multivariate demand distribution to a multivariate normal distribution that has mean zero and a covariance matrix that is the identity matrix. Given that this transformation is linear, the resource hyperplanes are transformed into new hyperplanes. This transformation has the property that the transformed hyperplanes are clustered into groups of hyperplanes that are approximately parallel.
The second linear transformation identifies a minimum orthogonal set of hyperplanes that spans the preceding transformed hyperplanes. The transformation uses factor analysis to identify this minimum spanning set.
The final transformation orthogonally rotates the minimum spanning set of hyperplanes so that they align along the coordinate axes. The linear, polynomial, or exponential value functions are transformed into similar functions after an orthogonal rotation of the coordinate axes. The multivariate normal distribution thus transformed has zero mean and identity covariance matrix, and is invariant to orthonormal rotational transformations. Hence, after the third transformation, the expected value function can be factored into a sum of products of univariate integrals, each with a closed form solution.
According to another aspect of the present invention, the solution is derivable when resource consumption follows a general rational model (e.g. the level of production refinement is proportional to the product of supporting resource allocations, wherein each item in the product enters with some positive or negative exponent).
Still another aspect provides for nonlinear elasticity in the value function. For instance, specific examples of terms in the value function (in a microeconomic model of product and component manufacture) might include product revenue. This term can be linear or nonlinear with coefficients that depend explicitly on product demand, the thereby reflect nonlinear elasticity.
Still another aspect provides that for a linear (also called general rational) resource consumption model, the refinement demand distribution can be inverted to yield a resource demand distribution that is also multivariate normal. From that inversion, an expectation value can be computed that depends explicitly on the resources. For example, the value function might include resource erosion (i.e. under-utilized resources at the end of a time period), or resource expediting (i.e. a need to expedite extra resources to fulfill refinement demand). Because refinement demand is uncertainxe2x80x94with probability distribution captured through a multivariate normalxe2x80x94resource demand is also uncertain with a probability distribution also captured through a multivariate normal that is obtained by effectively inverting the consumption model. Thus, the method for finding the expectation of the preceding resource based value function is directly applicable.
According to yet another aspect, the present invention can account for substitution of resources. Substitutability of resources in the production of a refinement suggests that certain resources can be substituted for other resources in the production model, but often with an incurred penalty, or cost. The solutions offered by the present invention apply directly to a general model of resource substitution with associated substitution costs.
Yet another aspect of the present invention provides for sensitivity analysis. The closed form solution of the expected value functionxe2x80x94i.e. the expectation of the value function over the refinement demand distributionxe2x80x94can be used to perform sensitivity analysis on each parameter in the value function.
Still another aspect of the present invention provides for sensor elements to be strategically located along data flows. These sensors might have embedded (or associated therewith) a probabilistic model that dynamically changes with the flow or update of various data through the sensor. The refined (up-to-date) probabilities can then be applied to the formation of the expected value function, and the solution thereof according to the present invention.